Lesson 3
MULTIPLY WHOLE NUMBERS
MULTIPLY DECIMALS
In this Lesson, we will answer the following:
1.
2.
3.
4.
HOW DO WE MULTIPLY BY A SINGLE DIGIT?
IN WRITING, HOW DO WE MULTIPLY WHOLE NUMBERS?
HOW DO WE MULTIPLY DECIMALS?
HOW CAN WE FIND THE AREA OF A RECTANGLE?
Let us face facts. The student will quickly replace written multiplication
with a calculator. (What if you don't have a calculator? What if you
don't have a pencil ) Therefore, what should the student know about
written multiplication that deserves to be called educational? The
student should know that it is based on the distributive property. The
student should also know the basics of placing the decimal point.
1. How do we multiply by a single digit?
Align the multiplier (on the bottom) with the ones
digit of the multiplicand, and draw a line. Then
multiply each digit of the multiplicand. Write the ones
digit of each product below the line. If there is a tens
digit, carry it -- add it -- to the next product.
"7 times 8 is 56." Write 6, carry 5.
"7 times 2 is 14, plus 5 is 19." Write 9, carry 1.
"7 times 6 is 42, plus 1 is 43." Write 43.
We can analyze this as follows:
7 has been distributed to each unit of 628: to
6 hundreds + 2 tens + 8 ones.
On the left (but compare the right):
7 × 8 ones = 56 ones, or simply 56. Write 6 below the line and
carry the 5 onto the tens column, because the 5 is 5 tens.
Next: 7 × 2 tens = 14 tens, plus 5 are 19 tens. Write 9 and carry
the 1 onto the hundreds column -- because 19 tens = 190. The
carried 1 is 1 hundred.
Finally: 7 × 6 hundreds = 42 hundreds, plus 1 is 43 hundreds.
Write 43.
When the multiplier has more than one digit --
-- follow the same procedure for each digit. However, when we
multiply by 5 tens, the product is 3140 tens. Therefore we write 0
in the tens column.
When we multiply by 2 hundreds, the product is 1256 hundreds,
and so we write 6 in the hundreds column.
2. In writing, how do we multiply whole numbers?
Write the multiplier under the multiplicand and draw
a line. Multiply the multiplicand by each digit of the
multiplier. Place the ones digit of each partial
product in the same column as the multiplying digit.
Then add the partial products.
Anticipating the next Question, if there were decimal points --
-- the multiplication would proceed in exactly the same way. In
the answer, we would then separate as many decimal places as
there are in the two numbers together; in this case, three.
Example 1.
0's within the multiplier.
907
× 308
7256
272156
279356
8 × 907 = 7200 + 56 = 7256
0 × 907 = 0
3 × 907 = 2700 + 21 = 272150
On distributing 8 ones, write 6 in the ones column.
Any number times 0 is 0, therefore it is not necessary to write any
digit in the tens column.
On distributing 3 hundreds, write 1 in the hundreds column.
It is not necessary to write rows of 0's. They add nothing to the
product.
3.
How do we multiply decimals?
.2 × 6.03
Ignore the decimal points -- do not align them -and multiply the numbers as whole numbers.
Then, starting from the right of the product,
separate as many decimal places as there are in
the two numbers together.
Example 2.
.2 × 6.03
Solution. Ignore the decimal points. Simply multiply
2 × 603 = 1206
Now we must put back the decimal points. Together, .2 and 6.03
have three decimal places. Therefore, starting from the right,
separate three places:
1.206
When we ignore a decimal point, we have in effect moved the
point to the right:
6.03 → 603
We have multiplied by a power of 10.
Therefore, to compensate and name the right answer, we must
divide by that power, we must separate the same number of
decimal places.
Example 3.
Solution.
.03 × .002
Ignore the decimal points.
3×2=6
Together, .03 and .002 have five decimal places. Therefore,
separate five places:
.00006
Example 4.
200 × .012
Solution. Ignore the decimal point. Multiply
200 × 12 = 2400
Again, to multiply whole numbers that end in 0's, first ignore the
0's, then replace them. But replace only the 0's on the end of
whole numbers. Do not replace the 0 of .012
Now separate three decimal places (.012):
2.400 = 2.4
These are simple problems that do not require a calculator
4. How can we find the area of a rectangle?
What is "1 square foot"?
1 square foot is a square figure in which each side is 1 foot.
We abbreviate "1 square foot" as 1 ft².
Now here is a rectangle whose base is 3 cm and whose height is 2
cm.
What do we call the small shaded square?
Since each side is 1 cm, we call it "1 square centimeter." And we
can see that the entire figure is made up 2 × 3 or 6 of them!
In other words, the area of that rectangle -- the space enclosed by
the boundary -- is 6 square centimeters: 6 cm².
If the rectangle were 3 by 3 -- that is, if it were a square -- then it
would be made up of 9 cm². If it were 3 by 4, the area would be
12 cm². And so on. In every case, to calculate the area of a
rectangle, we multiply the base times the height.
Area = Base × Height
When the length is measured in centimeters, the area is measured
in square centimeters: cm². And similarly for any unit of length.
We have illustrated this with whole numbers, but it will be true for
any numbers.
If the base is 12 in, and the height is 6.5 in, then to find the area,
multiply
12 × 6.5.
Now,
12 × 65 = 650 + 130 = 780
Therefore on separating one decimal place (6.5):
Area = 78 in²